Order-of-addition mixture experiments

Abstract

In a mixture experiment, m components are mixed to produce a response. The total amount of the mixture is a constant. Existing literature on mixture designs ignores the order of addition of the mixture components. This paper considers the Order-of-Addition (OofA) mixture experiment, where the response depends on both the mixture proportions of components and their order of addition. Empirical study demonstrates that if mixture-order interactions exist, then the optimal mixture proportions identified by traditional models may be misleading. Full Mixture OofA designs are created which ensure orthogonality between mixture model terms and addition order effects. These designs allow for the estimation of (1) typical mixture model parameters and (2) order-of-addition effects. Moreover, models which include both main effects and key mixture-order interactions are introduced.

Keywords:

• Design of experiments
• pairwise ordering model
• response surface design

Acknowledgments

We thank the reviewers for their constructive and helpful comments, which have led to insightful revisions. This paper is in honor of John A. Cornell, whose research (notably in mixture experiments) has a profound impact on our career. John Cornell is our teacher, mentor, and model. John passed away in 2016. The authors also thank Douglas Montgomery for his encouragement for working on such an important research subject.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplementary materials

MixOofAR.zip: A .zip file containing the modified Fish Patty data and two R files that are used to produce Tables 2–5 and Figures 1–3.

Appendix A: Proofs of some theoretical results for the designs presented in Section 2.

Appendix B: The original data from Cornell’s fish patty experiment, and another example using Food Science data (Aidoo, Afoakwa, and Dewettinck 2014).

Appendix C: Additional simulation results from Section 4.

Additional information

Funding

Dennis Lin is supported by the U.S. National Science Foundation, via Grant #DMS-18102925.

Notes on contributors

Nicholas Rios

Nicholas Rios is a PhD candidate and graduate fellow in the Department of Statistics at The Pennsylvania State University. His email address is [email protected].

Dennis K. J. Lin

Dr. Lin is a Distinguished Professor and Head in the Department of Statistics at Purdue University. He is a fellow of ASQ (also fellows of IMS, RSS, ISI, and ASA). His email address is [email protected].